Find integers x, y such that 242x + 2240y = gcd(242,2240)

Question

Find integers $x, y$ such that $242x + 2240y = \gcd(242, 2240)$. I am not sure what to do. I found the $\gcd(242, 2240)=2$. Not sure where to go from there. Please help.

Answer 1

Since Extended Euclidean algorithm is unknown we proceed directly. Cancelling an obvious factor of $\,2\,$ reduces to $\,11^2 x + 1120 y = (11^2, 1120) = 1\,$ since $11$ is coprime to $112$ and $10$.

$${\rm mod}\ 121\!:\,\ y \equiv \dfrac{1}{1120}\equiv \dfrac{1}{31}\equiv \dfrac{4}{124}\equiv \dfrac{4}3\equiv\dfrac{-117}{3}\equiv -39$$

Finally $\ x = (1 - 1120y)/121 = 361$

Beware $ $ Modular fraction arithmetic is valid only for fractions with denominator coprime to the modulus. See here for further discussion.

Answer 2

First, find the GCF by applying the euclidean algorithm as follows:

\begin{align*} 2240 &= 242(9) + 62 &\quad&\mbox{(equivalently, $62=2240 - 242(9)$)}\\ 242 &= 62(3) + 56 &\quad&\mbox{(equivalently, $56=242 - 62(3)$)}\\ 62 &= 56(1) + 6 &\quad&\mbox{(equivalently, $6=62 - 56(1)$)}\\ 56 &= 6(9) + 2 &\quad&\mbox{(equivalently, $2=56 - 6(9)$)}\\ 6 &= 3(2) + 0 \end{align*}

So, we have the GCF, which is $2$. Now, we need to write $2$ as a linear combination of $242$ and $2240$. To do this, we start with the equation on the bottom of the far right, $2=56-6(9)$ and substitute backwards.

\begin{align*} 2&=56 - 9(62-56) = 10(56) - 9 (62)\\ 2&=10(242-3(62)) - 9(62) = 10(242) - 39(62)\\ 2&=10(242) - 39(2240 - 242(9)) = 361(242) - 39(2240)\\ \end{align*}

Therefore, $x=361$ and $y=-39$. Keep in mind that as you substitute backwards the goal is writing $242$ and $2240$ as a linear combination. I recommend looking up the euclidean algorithm on Wikipedia as suggested and also diophantine equations on Wolfram.